Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

0
votes
0answers
23 views

What makes a topological insulator topological?

I understand that a topological insulator is one with an insulating bulk and conducting surface but I don't understand why or how the topological part comes into it. All of the resources I've found ...
-1
votes
1answer
73 views

Does CMB rule out that the universe is infinite?

If the universe were infinite, the energy of the big bang would have been long dissipated, and very little or nothing would hit us. Does the fact that CMB still comes roughly the same from every ...
2
votes
1answer
98 views

Why is $\rm{Conf}(\mathbb{R}^{1,1}) = \rm{Diff}(S^1) \times \rm{Diff}(S^1)$ and not $ \rm{Diff}(\mathbb{R}) \times \rm{Diff}(\mathbb{R})$?

The Minkowski metric for $\mathbb{R}^{1,1}$ is $$ ds^2 = dt^2 - dx^2 = du dv $$ for coordinates $$ u = t + x \hspace{1cm} v = t - x $$ If you do any coordinate transformation that acts independently ...
1
vote
0answers
35 views

Spacetime manifolds of 1+1 d systems for writing TQFT partition functions

Are the spacetime manifolds of two unentangled systems disconnected? This arises in the context of thinking of an operator whose expectation value we wish to take by writing this quantity in terms ...
0
votes
0answers
24 views

Definition of closed, compact manifold and topological spaces [migrated]

This is a very basic question but I seem not to get a "simple" definition anywhere that is at the same time rigorous and clear. I probably understand basic definitions of topology, topological spaces, ...
3
votes
1answer
96 views

What's the meaning of “inequivalent quantizations”?

The notion "inequivalent quantizations" is regularly used when topological terms are discussed. From what I've gathered so far, "inequivalent quantizations" means that there are different quantum ...
1
vote
0answers
28 views

What happens to large gauge transformations in gauges different from the temporal gauge?

There are already several questions regarding the meaning and definition of large gauge transformations. Discussions of large gauge transformations typically only happen in the context of the ...
1
vote
0answers
47 views

Wave function as a section of a complex line bundle to do QM in polar coordinates

If you want to change the coordinates of a Wave function $\Psi$ in 2D QM from cartesian to polar coordinates in a naive way one encounters a problem, namely the (naively defined) radial momentum ...
1
vote
1answer
52 views

Is it mathematically impossible to incorporate the space curvature into the equations of motion and gravity? [duplicate]

Obviously, I haven't studied GR, I know no more than common knowledge. However, I'm wondering, is it impossible to develop a mathematical model based on flat space, in which the new equations of ...
2
votes
1answer
74 views

Lefschetz and Witten indices$.$

I couldn't help but notice a formal similarity between the Lefschetz index $$ \mathrm{ind}(f)=\sum_k (-1)^k\operatorname{tr}(f_*|H_k) $$ and the Witten index $$ Z=\operatorname{tr}((-1)^Fe^{-\beta H}) ...
1
vote
1answer
51 views

Is there a definition for a *geometric entropy*?

In statistical mechanics, entropy of a system is usually defined as a measure of the system's micro-state randomness, or as an averaged "surprise" of its micro-state: \begin{equation}\tag{1} S_{\text{...
-1
votes
0answers
33 views

What is the meaning of mixed partial in terms of Physics and Math/Geometry?

How we can understand the meaning of mixed partial. As we used in thermodynamics mixed partial for pressure and volume etc. Also what is the geometric interpretations of it like in slope and concavity ...
1
vote
1answer
47 views

Understanding Anyonic Exchange

In the book of "Introduction To Topological Quantum Computation" by Jiannis K. Pachos, in chapter 5, it tries to explain anyonic exchange. In the following, the $m$ and $e$ quasi-particles are ...
0
votes
2answers
61 views

Rigourous formalism of Hamiltonian mechanics on Manifolds

I'm looking for books / articles that provide rigorous formulations of Hamiltonian mechanics on Manifolds. I found the book "Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds" [1]...
0
votes
0answers
30 views

Reference for topology for topological insulators

In the field of topological insulators What topological space do they talk of? Looking for some resources that sheds light on the topology part of topological insulator
2
votes
1answer
50 views

2-sheeted Riemann surface with 2 branch cuts and Torus

A 2-Sheeted Riemann surface, with 2 branch cuts has a genus 1. A ring torus also has a genus 1 (In fact, section 13.4 of John Terning's book, modern supersymmetry and dynamics and duality claims that ...
6
votes
2answers
80 views

Particle on a circle with magnetic flux$.$

I am trying to understand the model studied in 1905.09315 §2, to wit, a $0+1$ dimensional theory with target space $\mathbb S^1$ with non-trivial magnetic flux: $$ \mathcal L=\frac12m\dot q^2-\frac{i}{...
4
votes
0answers
46 views

Pulling apart the atoms of a topological insulator

Consider a topological insulator. In order to destroy a topological phase, the band gap of the bulk system should close at some point (passing thru a conducting state), but if the atoms that make up ...
2
votes
0answers
44 views

Fluxes on a finite group $G$

So I've been studying about topological quantum computation and I have a few questions I haven't been able to solve. The first one is why fluxes take values on a finite group $G$? Does it have to do ...
3
votes
3answers
118 views

Rigorous procedure of gluing together two spacetimes

There seems to exist a procedure of "gluing two spacetimes together". In particular I've seem this mentioned in the context of gravitational collapse. The examples I've seem where that of gluing ...
2
votes
0answers
21 views

What is the shape of the universe? [duplicate]

If it's flat then how a volume can be flat? And I've read that it's actually not flat .....it's a "Poincaré dodecahedral space". Any suggestions for books posts or articles are highly appreciated.
3
votes
0answers
52 views

Does the Einstein field equation uniquely determines the topology of spacetime? [duplicate]

I am trying to understand whether the Einstein field equation uniquely determines the topology of spacetime. As far as I know, given a metric, we can always find the induced topology. However, I was ...
1
vote
0answers
54 views

String topology in string theory

How do string topology, string field theory and topological strings interact? Does anybody see a global picture? By string topology I mean the TQFT based on the homology of the space of loops ...
0
votes
0answers
35 views

Can the shape of our Universe be a Mobius strip? [duplicate]

The Friedmann Equations describe three possibilities for the shape of our using General Relativity, I read in a book that the shape of our Universe is a 3-sphere such that in any direction if you ...
1
vote
0answers
32 views

Critical points of vector field with zeros in the magnitude

I am studying a vector field which has critical points (sources, sinks, saddle points and centers). The magnitude of the vector field goes to zero smoothly in these points, however. Contrast that to ...
1
vote
1answer
82 views

Berry phase: Spin in a magnetic field parameter space manifold

Canonical example for Abelian Bery phase is a spin in a magnetic filed, e.g.. Usually authors calculate spin eigenstates, conclude that they don't depend on B in spherical components and so deduce ...
2
votes
1answer
63 views

Topological theta-term as a background electric/magnetic field?

The topological $\theta$-term in the Schwinger model (1+1-dimensional QED) can be interpreted as a background electric field, as explained in Chapter 7.1.2 of Tong's lecture notes. The same holds true ...
2
votes
1answer
88 views

Theta-dependence of massive Schwinger model

I've read in Coleman's paper on the massive Schwinger model (and in other papers on the same topic, like this one) that the model's Hamiltonian contains a topological $\theta$-term. However, if I ...
1
vote
1answer
113 views

Black holes in p-adic gravity/ultra-metric metric field? [closed]

As a radically different to beyond standard general relativity, at least from the type of geometry it deals with: Consider p-adic gravity and/or general relativity defined on certain ultra-metric ...
0
votes
0answers
9 views

What’s the topology of critical region?

Duhem said the aim of physics is natural classification. I think topology and geometry are a wonderful way to link analogous parts among different phenomena. Thus we can classify and predict facts. ...
5
votes
1answer
185 views

“Hidden” theta-term in Hamiltonian formulation of Yang-Mills theory

I've read in David Tong's lecture notes on gauge theory that the Hamiltonian of Yang-Mills theory does not depend on the angular parameter $\theta$, because it can be absorbed in the electric field: $...
1
vote
0answers
36 views

Symmetry operators of a Bloch Hamiltonian

Consider a lattice with a 3 atom basis, e.g. the Lieb lattice, and some completely arbitrary on-site energies and hopping energies and phases between the different atoms. In momentum space we can ...
2
votes
1answer
39 views

Change in area on a 3-sphere bounded by a trajectory due to a differential change in trajectory

I have a 3 dimensional spherical topology, and I draw a curve onto the sphere labelled by $\vec{n}(\vec{r},t)$. The area bounded by the curve is termed the "Wess Zumino Action" (Hence my motivation to ...
2
votes
0answers
49 views

Hausdorff property in Minkowski spacetime

In the 4-dimensional Minkowski spacetime, for a given point $x = (x^0,x^1,x^2,x^3)$, its timelike future/past set is defined as, $$ I^{\pm}(x) = \{y =(y^0,...,y^3) \in \mathbb{R}^4 : \eta_{\mu \nu}(y-...
0
votes
0answers
12 views

For the torus to rotate 180 degrees around the East-West symmetry axis, what happens?

(Suppose to ignore the deformed friction and torus when rotating)
3
votes
1answer
127 views

Why doesn't the $\theta$ Angle Renormalize?

The $\theta$ term for Yang-Mills takes the form $$L_{\theta}=\frac{\theta}{64\pi^2}\varepsilon^{\mu\nu\rho\sigma}F^a_{{\mu\nu}}F^a_{\rho\sigma}$$ A fact that I have heard is that $\theta$ does not ...
0
votes
0answers
22 views

Inside a shell full of strings, what are the components of the network, and what the possible defects?

Imagine that we have a (very) large shell full of holes, and flexible strings go (randomly) from one hole to another. Inside the shell, there will be a network of strings. Can one split this network ...
0
votes
1answer
34 views

Weyl Hamiltonian - Monopole in momentum space

Consider the Weyl Hamiltonian in momentum space: $$ \mathcal{H} = \hbar v_F \chi (\bf{\sigma}\cdot{\bf k}) , $$ where $\sigma^i$ are the Pauli matrices and $\chi$ the chirality of the Weyl node. ...
2
votes
1answer
45 views

Classical field theory with fields on different base spaces

Keeping things at a "basic level", a field is a function from a base manifold (of dimension D) to some other space. Usually the base manifold is the spacetime but may be something different (a lattice,...
1
vote
2answers
94 views

Why are systems in classical mechanics sets and not vector spaces?

One of my professors told me that a state in a classical system is an element of a set, while in quantum it's an element of a vector space. From that, we combine systems in classical physics using a ...
0
votes
2answers
69 views

Triangle inequality and the pseudo-metric of general relativity

The pseudo-metric, such as those used in general relativity, can be expressed as the following sum: $$ ds^2=\sum_{\nu}\sum_{mu}g_{\nu\mu}dX_\nu dX_{\mu} $$ The elements $g_{\nu\mu}$ can be organized ...
1
vote
1answer
73 views

Parameter space of $SO(3)$ and $SU(2)$

Is it parameter space of $SO(3)$ and $SU(2)$ are same? can we use quaternions to represent both groups? what about their connectedness?
1
vote
1answer
160 views

Basics of topological order and its relation to entanglement

What is a topological order that drives a topological phase transition? How is it different from say magnetic ordering or the superfluid ordering? What is its relation with entanglement? Please ...
0
votes
0answers
38 views

IR divergence Feynman diagram topology query

I am trying to calculate the superficial degree of Infrared divergence. To do this I am reading section 12 of this source. It seems you can calculate it by a method involving the 'shrinking' of ...
0
votes
0answers
16 views

What does the interferogram in Michelson interferometer mean about the topology of the specimen?

I'm want to know how can I evaluate the surface of a fibre by using the interferometer of Michelson. The object beam in this method is reflected from the object and combined with the reference beam to ...
1
vote
0answers
97 views

Isn't mass from pure topology in the absence of matter a contradiction?

Consider the 3 dimensional projective space minus the infinity point, and empty of matter. We see that it has ADM mass. In other words, a perfectly fine geometry, orientable and asymptomatically flat,...
0
votes
1answer
39 views

Can any body suggest me a set up that I can build it up to study the surface roughness of transparent polymeric fibres? [closed]

I need to study the surface of nylon-6 blank fibres with a thickness of nearly 40 micrometre. I work in optics lab with many optical components such as beam splitters, lenses, mirrors, etc; can I ...
0
votes
0answers
16 views

Can I build up a profilometry set up, to study the surface roughness of polymer fibres?

I need to study the surface roughness of some fibres, I was told that the best way is the profilometer, but can I design and build up a set up of a profilometer?
0
votes
0answers
10 views

How interferometric techniques can be used to investigate the topology of polymers?

I need to study the topography of a specific polymer, knowing that my lab is an optics lab with many interferometric techniques, how can I use these techniques to investigate the topology?
0
votes
0answers
14 views